%I
%S 1,3,10,32,101,318,1003,3173,10071,32071,102453,328260,1054620,
%T 3396757,10965653,35475159,114989969,373400210,1214529314,3956450250,
%U 12906762704,42159475998,137877383739,451403471067,1479329370617,4852295325254,15928202158814,52321416289743,171966242037941,565480887258368,1860228812665716,6121446895971437
%N The sum of the heights of all bargraphs of semiperimeter n (n>=2).
%D A. Blecher, C. Brennan, A. Knopfmacher, The height and width of bargraphs, Discrete Appl. Math., 180, 2015, 3644 (see pp. 4142).
%F a(n) = Sum(k*A278132(n,k), k>=0).
%e a(4)=10; indeed, the bargraphs of semiperimeter 4 correspond to the compositions [3],[1,2],[2,2],[2,1],[1,1,1] and the sum of their heights is 3+2+2+2+1=10.
%p x := z: y := z: eq := G(h) = x*(y+G(h))+y*G(h1)+x*(y+G(h))*G(h1): ic := G(1) = x*y/(1x): sol := simplify(rsolve({eq, ic}, G(h))): for j to 17 do g[j] := factor(simplify(rationalize(simplify(subs(h = j, sol))))) end do: H[1] := x*y/(1x): for j from 2 to 50 do H[j] := factor(g[j]g[j1]) end do: for j to 17 do Hser[j] := series(H[j], z = 0, 50) end do: T := proc (n, k) coeff(Hser[k], z, n) end proc: seq(add(k*T(n, k), k = 1 .. n1), n = 2 .. 45);
%Y Cf. A278132.
%K nonn
%O 2,2
%A _Emeric Deutsch_, Dec 31 2016
